Why Update a Parameter Distribution?
Therapeutic drug monitoring gives information about an individual, but a measured concentration is not itself a pharmacokinetic parameter. The interpretation depends on the dose and sampling history, the PK model, measurement uncertainty, and what was plausible about the individual before the measurement. Bayesian updating provides a formal way to combine that prior information with the information in the individual’s concentration data.
The result is not just a revised value for clearance, volume, or another parameter. It is a posterior distribution: a probability distribution describing uncertainty about the parameter after considering the data and the assumptions used to interpret them. A posterior can support a point summary when one is needed, but retaining its uncertainty matters for forecasts and dose decisions.
Bayes’ Rule for Individual Parameters
For a continuous parameter, Bayes’ rule states that the posterior density is proportional to the prior density multiplied by the likelihood. A normalizing constant makes the posterior density integrate to one:
The variable \(u\) in the denominator ranges over the possible values of the parameter. The denominator is the marginal probability density of the observed data under the prior and model; it does not depend on \(\theta\). For a discrete parameter with possible values \(\theta_j\), replace the integral with a sum over those values.
The likelihood is a function of the candidate parameter once the observed data are fixed. It is not, by itself, a probability distribution over parameter values. The prior supplies that distribution before the data; the posterior follows from their product and normalization. A prior can encode information from previous knowledge or analysis, but it must be specified on the same parameter scale and for the same individual parameter being updated.
Bayesian updating relies on substantive assumptions. The prior must represent the intended pre-data uncertainty, and the likelihood must be appropriate for the dose, sampling times, PK model, and measurement process. The data must not be counted twice—for example, by using the same concentration both to construct the prior and again as new evidence. The parameter should also have a coherent interpretation over the observation period; if it changes during that period, a single fixed parameter may not describe the data.
A Normal-Normal Update
A useful calculation has a normal prior for a scalar parameter and a normal likelihood for an estimate of that parameter with known standard error. This is an illustrative summary-data setup: assume the concentration analysis has already supplied an estimate \(z\), with \(z\mid\theta\sim N(\theta,s_y^2)\). This tutorial takes that likelihood as given rather than deriving it from a nonlinear concentration model.
Suppose the prior is \(\theta\sim N(\mu_0,s_0^2)\). Multiplying the normal prior density by the normal likelihood gives a normal posterior. Its variance and mean are:
The inverse variances are precisions. Thus, in this special case, prior and data precisions add. The posterior mean is a precision-weighted average: the estimate with smaller variance receives more weight. These formulae require the stated normal distributions and known variances; they are not universal formulae for arbitrary PK likelihoods or priors.
Worked Example: Updating an Elimination-Rate Parameter
Consider a hypothetical individual whose elimination-rate parameter is \(k\), measured in \(\text{h}^{-1}\). Suppose prior information is represented by \(k\sim N(0.100,0.030^2)\). An analysis of that individual’s concentration data is summarized by \(z=0.140\ \text{h}^{-1}\), with known standard error \(0.020\ \text{h}^{-1}\), and suppose \(z\mid k\sim N(k,0.020^2)\). Assume the prior and this new data source are independent conditional on \(k\), and that the summary likelihood is appropriate.
The posterior variance is the reciprocal of the sum of the prior and data precisions:
The posterior mean is:
The posterior standard deviation is \(\sqrt{0.0002769}\approx0.01664\ \text{h}^{-1}\). The posterior mean lies between the prior mean and the data estimate, and is closer to \(z\), whose standard error is smaller than the prior standard deviation. Under the normal posterior model, an approximate 95% credible interval is \(0.1277\pm1.96(0.01664)\), or approximately \(0.0951\) to \(0.1603\ \text{h}^{-1}\). Conditional on the specified prior and likelihood, this interval contains 95% of the posterior probability for \(k\). It is not a statement that the fixed parameter has a 95% probability of lying in this particular interval under a frequentist interpretation.
A normal distribution technically permits negative \(k\), although an elimination rate is positive. Here the prior places only a very small probability below zero, and the posterior probability below zero is smaller still. The normal approximation is therefore a reasonable illustration for these numbers. If meaningful posterior probability lay outside the physically valid parameter space, a positive-support model or an appropriate transformation would be needed.
Why the Normal Update Works
The normal-normal result follows by collecting the terms in the exponent of the prior density times the likelihood. Ignoring factors that do not depend on \(\theta\), the product is proportional to:
Expand the two squared terms. The coefficient of \(\theta^2\) inside the brackets is \(1/s_0^2+1/s_y^2\), and the coefficient of \(-2\theta\) is \(\mu_0/s_0^2+z/s_y^2\). Completing the square therefore gives a normal kernel with variance \(s_1^2=(1/s_0^2+1/s_y^2)^{-1}\) and mean \(s_1^2(\mu_0/s_0^2+z/s_y^2)\). The remaining terms do not depend on \(\theta\) and are absorbed into the normalizing constant. This verifies both update formulae.
The derivation also explains why greater precision yields greater influence: each source contributes a quadratic term to the exponent, and its coefficient is the corresponding precision. If the likelihood is asymmetric, has multiple modes, or is paired with a non-normal prior, this completed-square argument does not apply; Bayes’ rule itself still applies, but the posterior need not be normal.
Worked Example: Updating With a Second Independent Observation
Continue with the hypothetical prior \(k\sim N(0.100,0.030^2)\), but now suppose there are two independent summary estimates: \(z_1=0.140\ \text{h}^{-1}\) with standard error \(0.020\ \text{h}^{-1}\), and \(z_2=0.120\ \text{h}^{-1}\) with standard error \(0.030\ \text{h}^{-1}\). Assume each estimate has a normal likelihood centered on \(k\), and that they are conditionally independent given \(k\).
The prior and both likelihoods multiply. Thus the total precision is the sum of the three precisions, while the posterior mean is the sum of the precision-weighted values divided by total precision:
The posterior standard deviation is about \(0.01455\ \text{h}^{-1}\), less than in the one-observation example. This is expected because an additional independent observation has added information. Sequential updating gives the same result as using both observations at once, provided the first posterior is used as the second step’s prior and only the new observation is used in that step. Reusing \(z_1\) as new data after it has already contributed to the first posterior would count it twice and make uncertainty too small.
Bayes’ Rule Beyond the Normal Case
Bayes’ rule is not restricted to continuous normal distributions. For a parameter with a small set of possible values, the posterior probability of each value is its prior probability multiplied by the likelihood for the observed data, then divided by the sum of these products. The next example illustrates the calculation; the likelihood values are hypothetical and are assumed to come from an appropriate model for the observation.
Worked Example: Updating Two Possible Clearance Ranges
Suppose, for illustration, an individual’s clearance is represented by two categories: lower clearance \(L\) or higher clearance \(H\). Before new data, let \(P(L)=0.70\) and \(P(H)=0.30\). Suppose the observed concentration result \(y\) has likelihoods \(p(y\mid L)=0.20\) and \(p(y\mid H)=0.60\), expressed in the same density units. The larger likelihood under \(H\) means the observed result is more compatible with that category under the assumed model; it does not alone determine the posterior.
The unnormalized posterior weights are \(0.20(0.70)=0.14\) for \(L\) and \(0.60(0.30)=0.18\) for \(H\). Their sum is \(0.32\), so:
The data shift the probability toward higher clearance, but the posterior still reflects the prior: the prior probability of \(H\) was only 0.30, and the likelihood advantage is not large enough to make \(H\) certain. In a real analysis, clearance would generally be represented more richly than two categories; this calculation illustrates normalization and the role of prior probabilities.
Assumptions, Checks, and Alternatives
Before relying on an individual posterior, check that the information entering the update matches the clinical record and the intended model. In particular, confirm dose and sampling times, units, the relevant dosing history, and whether the measured sample is interpretable. The earlier material on analytical uncertainty and contaminated samples is pertinent: a transcription error, assay issue, or mistimed sample can make an apparently precise likelihood misleading. Also check whether a parameter can reasonably be treated as stable over the period represented by the data.
A posterior that appears implausibly concentrated or far from the prior deserves investigation rather than automatic acceptance. Possible causes include a poorly calibrated prior, an incorrect observation model, unaccounted variability, dependence between observations treated as independence, or a data error. Prior-data conflict can be assessed by comparing the observed data’s compatibility with prior-plausible parameter values and by examining whether reasonable changes in the prior substantially change the posterior. These checks do not prove the model is correct, but can reveal conclusions that depend strongly on questionable assumptions.
| Approach | What it uses | What it provides and what it misses |
|---|---|---|
| Bayesian updating | A prior distribution and a likelihood | A posterior distribution that combines both sources and represents parameter uncertainty, conditional on the assumptions. |
| Likelihood-based fitting | The likelihood from the observed data | A data-based estimate and uncertainty assessment, without incorporating a prior distribution in the same way. |
| Prior-only estimate | Prior information, without the individual’s new data | A starting distribution or summary, but no update from the current individual observations. |
These approaches answer related but not identical questions. A Bayesian posterior is conditional on both the prior and the likelihood; it is not automatically more reliable merely because it combines two sources. A likelihood-only analysis can be useful when one wants the data contribution without prior weighting, while a prior-only value cannot reflect new concentration evidence. In sparse data settings, the prior may have substantial influence, so reporting only a posterior point estimate can conceal important uncertainty.
Common Mistakes and Practical Limits
- Treating the likelihood as a posterior. The likelihood describes relative compatibility of data with parameter values; it becomes a posterior only after multiplication by a prior and normalization.
- Assuming every update is a weighted average. The simple precision-weighted formula requires a normal prior and normal likelihood with the stated variance assumptions. Other combinations can yield skewed or multimodal posteriors.
- Counting observations twice. Each observation should enter the update once. Reusing data already represented in a prior or posterior artificially increases apparent information.
- Ignoring parameter constraints. A mathematical distribution may assign probability to impossible values. Check its support and use an appropriate model when that probability is material.
- Reporting a posterior mean without context. State the parameter, units, prior and model assumptions, and an uncertainty summary. A point estimate alone does not show how strongly the data updated the prior.
Bayesian updating is useful because it makes the combination of prior knowledge and new individual evidence explicit. Its limitation is equally important: the posterior is conditional on the chosen prior and likelihood. It cannot repair an invalid model, incorrect sampling information, or unreliable data. A posterior distribution is therefore an input to clinical interpretation and forecasting, not a substitute for checking whether the evidence and assumptions are credible.
Check Your Understanding
Try each question, then compare your reasoning with the solutions.
- In Bayes’ rule for a continuous PK parameter, what roles do the prior, likelihood, and normalizing constant play?
- A normal prior for a parameter has variance \(0.04\), and a normal data summary has variance \(0.01\). Which source receives more weight in the normal-normal posterior mean, and why?
- Why does adding a second independent observation usually reduce posterior variance in the normal-normal example?
- A likelihood is much larger for one parameter value than another. Does that alone establish which value has the greater posterior probability? Explain.
- Give one reason a posterior may be misleading even when the Bayes’ rule calculation is arithmetically correct.
Solutions. (1) The prior represents parameter uncertainty before the data; the likelihood measures how compatible the data are with each parameter value; the normalizing constant ensures the posterior integrates to one. (2) The data summary receives more weight because its variance is smaller and its precision, the inverse variance, is larger: \(1/0.01=100\), compared with prior precision \(1/0.04=25\). (3) Under conditional independence, the second observation contributes additional likelihood precision, so total precision increases and its reciprocal, posterior variance, decreases. (4) No. Posterior probability depends on both likelihood and prior probability, followed by normalization. (5) For example, the likelihood may be wrong because sampling time or dose history was recorded incorrectly, so the mathematically correct update answers the wrong model-based question.